This Demonstration considers the quantum-mechanical system of a free electron in a constant magnetic field, with definite values of the linear and angular momentum in the direction of the field. The wavefunction is plotted in a plane normal to the magnetic field. The corresponding energies are the equally spaced Landau levels, similar to the energies of a harmonic oscillator. These results find application in the theory of the quantum Hall effects.

You can select a 3D plot of the wavefunction, a plot of the radial function or an energy-level diagram. The first slider varies the magnetic field strength . You can then select and , the radial and angular quantum numbers, respectively.

Details

The nonrelativistic Hamiltonian for an electron in a magnetic field , where is vector potential, is given by

,

where and are the mass and charge of the electron, respectively. We also make use of the Coulomb gauge condition .

For a constant field in the direction, , it is convenient to work in cylindrical coordinates, . With a convenient choice of gauge, the vector potential can be represented by

,

.

This gives

.

The Schrödinger equation for is given by

.

The equation is separable in cylindrical coordinates, and we can write

,

for definite values of the angular and linear momenta. We consider only angular momentum anticlockwise about the axis. We set and consider only motion in a plane perpendicular to the magnetic field. Introducing atomic units , the radial equation reduces to

.

The solution with the correct boundary conditions as is given by

,

where is an associated Laguerre polynomial. The corresponding energy eigenvalues are

.

These are the well-known Landau levels, which are equivalent to the levels of a two-dimensional harmonic oscillator with